A174965 Length of the n-th run of consecutive terms in A000961.

5, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,1

Comments

Let b(n) be partial sums of this sequence, as b(0) = 0, b(n) = b(n-1) + a(n). Then 2's in this sequence occur at certain positions k so that A000961(b(k))-1 = one of the terms of A006549. E.g., a(6625) = 2, and b(6625) = 6637, with A000961(6637) = 65537, the end of 2-run (65536,65537) in A000961. (The start of that 2-run, 65536, is listed as A006549(12), and 12 = 6637-6625.) - Antti Karttunen, Jul 24 2017

Position of 2s: {5, 11, 36, 62, 1068, 6625, 12358, 43566, ...},

position of 3: {2}, position of 5: {1}. All other positions pertain to 1 for A000961(n) <= 10^8. - Michael De Vlieger, Jul 25 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..9987 (computed by using the 10000 term b-file of A000961 computed by T. D. Noe)

Example

n a(n) run in A000961
1:  5 (1 2 3 4 5)
2:  3 (7 8 9)
3:  1 (11)
4:  1 (13)
5:  2 (16 17)
6:  1 (19)
7:  1 (23)
8:  1 (25)
9:  1 (27)
10: 1 (29)
11: 2 (31 32)
12: 1 (37)
13: 1 (41)
14: 1 (43)
15: 1 (47)
16: 1 (49)
17: 1 (53)
18: 1 (59)
19: 1 (61)
20: 1 (64)
21: 1 (67)
22: 1 (71)
23: 1 (73)
24: 1 (79)
25: 1 (81)
26: 1 (83)
27: 1 (89)
28: 1 (97)
29: 1 (101)
30: 1 (103)
31: 1 (107)
32: 1 (109)
33: 1 (113)
34: 1 (121)
35: 1 (125)
36: 2 (127 128)
37: 1 (131)
38: 1 (137)
etc.

Mathematica

With[{n = 500}, Function[s, Function[t, Length /@ DeleteCases[Split@ ReplacePart[t, Map[# -> 1 &, s]], k_ /; First@ k == 0]]@ ConstantArray[0, Max@ s]]@ Sort@ Flatten[{{1}}~Join~Array[Function[p, Map[p^# &, Range@ Floor@ Log[p, n]] ][Prime@ #] &, PrimePi@ n]]] (* Michael De Vlieger, Jul 25 2017 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A174965 n) (let ((k (+ 1 (Apartsums_of_A174965 (- n 1))))) (let loop ((prev (A000961 k)) (i 1)) (let ((m (A000961 (+ k i)))) (if (= m (+ 1 prev)) (loop m (+ i 1)) i)))))
;; An auxiliary partial sums function not submitted separately:
(definec (Apartsums_of_A174965 n) (if (zero? n) n (+ (A174965 n) (Apartsums_of_A174965 (- n 1)))))
;; Antti Karttunen, Jul 24 2017

Crossrefs

Cf. A000961, A006549, A057820.

Sequence in context: A132199 A111142 A179626 * A159671 A094853 A096937

Adjacent sequences: A174962 A174963 A174964 * A174966 A174967 A174968

Keyword

nonn

Author

Pahikkala Jussi, Apr 02 2010

Extensions

Terms a(33) & a(36) corrected, more terms added by Antti Karttunen, Jul 24 2017

Status

approved